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5937Reservation Wage With Exponential OffersJob offers arrive sequentially and independently, each an Exponential random variable with rate 1 (mean 1). After each offer you accept it (and stop) or reject it forever and pay a search cost c = 0.2 to see the next; there is no deadline. Find the optimal stationary reservation level a above which you accept, and the expected wage you end up accepting under that policy.概率中等数值题未尝试免费5938Bird in the Hand vs Discounted WaitYou face two periods. In period 1 a reward X1 ~ Uniform(0,1) is offered; accept it now to receive X1, or wait. If you wait, in period 2 you must accept X2 ~ Uniform(0,1), but a reward received in period 2 is worth only a fraction beta = 0.8 of its face value (discounting). No recall. Find the optimal period-1 acceptance threshold and the expected payoff of the optimal policy.概率简单数值题未尝试免费5939Secretary Selection When Ties Are PossibleThree items arrive in uniformly random order. Their qualities are NOT all distinct: two of them have quality 2 (tied for best) and one has quality 1. After each item you observe its quality relative to those seen so far, reported as 'higher', 'tied', or 'lower' (so a tie is visible). You accept irrevocably or reject (the last is forced). You want to maximize the expected quality of the item you accept. Find the optimal policy and the maximum expected quality, and explain how the possibility of an observed tie changes what you can guarantee.概率中等数值题未尝试免费5940Secretary With an Unknown Number of CandidatesCandidates arrive one at a time in uniformly random order, but the TOTAL number N is itself random: N = 2 with probability 1/2 and N = 3 with probability 1/2, and you do not learn N in advance. After each arriving candidate you observe its rank relative to those seen so far and must irrevocably accept or pass; once the stream ends, if you never accepted you lose. You win only if the candidate you accept is the overall best of all N who arrived. Find the policy that maximizes the win probability and that probability.概率困难数值题未尝试面试订阅5941The 1/e Law of Best ChoiceIn the classic secretary problem with n candidates (relative ranks only, irrevocable choices), the look-then-leap rule observes the first r candidates without choosing and then accepts the first later candidate who beats all seen so far. For large n, write r = t*n and derive the limiting win probability as a function of the skip fraction t in (0,1). Then find the t that maximizes it and the resulting optimal asymptotic probability of selecting the single best candidate.概率中等derivation未尝试免费